If $I_n=\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cot ^n x d x$, then:

$\frac{1}{I_2+I_4}, \frac{1}{I_3+I_5}, \frac{1}{I_4+I_6}$ are in G.P

$I_2+I_4, I_3+I_5, I_4+I_6$ are in A.P

$I_2+I_4,\left(I_3+I_5\right)^2, I_4+I_6$ are in G.P

JEE MATHEMATICS MCQ: DEFINITE INTEGRATION

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Analyzing +4 | -1

I_n=\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cot ^n x d x

, then:

View answer and explanation

Correct option: D

\frac{1}{I_2+I_4}, \frac{1}{I_3+I_5}, \frac{1}{I_4+I_6}

are in A.P

Option D (

\frac{1}{I_2+I_4}, \frac{1}{I_3+I_5}, \frac{1}{I_4+I_6}

are in A.P) is correct because by using integration by parts, we can find a recurrence relation for

I_n

and show that the reciprocals are in arithmetic progression.

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